Average Error: 33.6 → 10.8
Time: 8.1s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.1192253259106201 \cdot 10^{-138}:\\ \;\;\;\;-0.5 \cdot \frac{1}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 2.818024375880192 \cdot 10^{+91}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a}\right)\\ \end{array} \]
\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -1.1192253259106201 \cdot 10^{-138}:\\
\;\;\;\;-0.5 \cdot \frac{1}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\

\mathbf{elif}\;b \leq 2.818024375880192 \cdot 10^{+91}:\\
\;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a}\right)\\


\end{array}

(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1192253259106201e-138)
   (* -0.5 (/ 1.0 (* 0.5 (- (/ b c) (/ a b)))))
   (if (<= b 2.818024375880192e+91)
     (* -0.5 (/ (+ b (sqrt (fma a (* c -4.0) (* b b)))) a))
     (* -0.5 (* 2.0 (/ b a))))))
double code(double a, double b, double c) {
	return (-b - sqrt((b * b) - (4.0 * (a * c)))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1192253259106201e-138) {
		tmp = -0.5 * (1.0 / (0.5 * ((b / c) - (a / b))));
	} else if (b <= 2.818024375880192e+91) {
		tmp = -0.5 * ((b + sqrt(fma(a, (c * -4.0), (b * b)))) / a);
	} else {
		tmp = -0.5 * (2.0 * (b / a));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.3
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.1192253259106201e-138

    1. Initial program 49.9

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Simplified49.8

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]

    3. Applied egg49.8

      \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]

    4. Taylor expanded in b around -inf 13.1

      \[\leadsto -0.5 \cdot \frac{1}{\color{blue}{0.5 \cdot \frac{b}{c} - 0.5 \cdot \frac{a}{b}}} \]

    5. Simplified13.1

      \[\leadsto -0.5 \cdot \frac{1}{\color{blue}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}} \]

    if -1.1192253259106201e-138 < b < 2.8180243758801921e91

    1. Initial program 11.1

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Simplified11.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]

    if 2.8180243758801921e91 < b

    1. Initial program 44.4

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Simplified44.4

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]

    3. Applied egg44.5

      \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}} \]

    4. Taylor expanded in a around 0 4.0

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{b}{a}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1192253259106201 \cdot 10^{-138}:\\ \;\;\;\;-0.5 \cdot \frac{1}{0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \leq 2.818024375880192 \cdot 10^{+91}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))